We conjecture a geometrical form of the Paley–Wiener theorem for the Dunkl transform and prove three instances thereof, by using a reduction to the one-dimensional even case, shift operators, and a limit transition from Opdam’s results for the graded Hecke algebra, respectively. These Paley– Wiener theorems are used to extend Dunkl’s intertwining operator to arbitrary smooth functions. Furtherm...

where x̄ is the complex conjugate of x This paper uses the machinery of almost periodic functions to prove that even without uniform convergence the connection between a pair of almost periodic functions and the constants of the associated Fourier series exists for both the convolution and cross-correlation. The general results for two almost periodic functions are narrowed and applied to Ramanu...

1. Non-Banach limits C(R), C∞(R) of Banach spaces C[a, b] 2. Banach completion C o (R) of C c (R) 3. Rapid-decay functions, Schwartz functions 4. Non-Fréchet colimit C∞ c (R) of Fréchet spaces 5. LF-spaces of moderate-growth functions 6. Strong operator topology 7. Generalized functions (distributions) on R 8. Tempered distributions and Fourier transforms on R 9. Test functions and Paley-Wiener...

We point out that an approximation property of Gaussian functions, derived in a recent work, is a direct corollary to the work of Wiener on the closure of translations in L1 and L2. This observation not only simplifies the proof of the approximation property, but also renders the result applicable, in a more general setting, to other functions (not necessarily Gaussian).

Information about the tuning and timing of excitation in cochlear axons with low-characteristic frequency (CF) is embodied in the first-order Wiener kernel, or reverse correlation function. For high-CF axons, the highest-ranking eigenvector (or singular vector) of the second-order Wiener kernel often can serve as a surrogate for the first-order kernel, providing the same information. For mid-CF...

New type Paley–Wiener theorems for the modified multidimensional Mellin and inverse Mellin transforms are established. The supports of functions are described in terms of their modified Mellin (or inverse Mellin) transform without passing to the complexification.

We present a uniied approach to error estimates of periodic interpolation on equidistant, full, and sparse grids for functions from a scale of function spaces which includes L 2 {Sobolev spaces, the Wiener algebra and the Korobov spaces.

This is a concise survey of some results and open problems concerning Wiener-Hopf factorization and almost periodic factorization of matrix functions. Several classes of discontinuous matrix functions are considered. Also sketched is the abstract framework which unifies the two types of factorization. Mathematics Subject Classification (2010). Primary 47A68; Secondary 30E25, 43A75, 45F15, 47B35 .